Derivatives
Six units on the derivative, the slope of a curve at a single point. You define it as a limit of secant slopes, read it as a new function with its own graph, learn the rules that make computing it quick, use it for velocity and acceleration, and finish by finding the highest and lowest points of a graph. The calculus lab lets you shrink a secant into a tangent and drag a tangent line along a curve while the slope updates.
About this course
Of all rectangles with a perimeter of 40, the one with the largest area is the 10 by 10 square. A rock dropped from 64 feet hits the ground at t = 2 moving at 64 feet per second. Both answers come from the same idea: the derivative, the slope of a curve at a single point. The course builds it from the ground up, starting with the secant from x = 1 to x = 1 + h on f(x) = x², whose slope is exactly 2 + h, so that letting h shrink leaves 2.
The six units go from definition to use. Defining the derivative comes first, then the derivative as a function: for x², one formula f′(x) = 2x gives −6 at x = −3, 0 at the bottom and 6 at x = 3. The power rule follows, so the derivative of x¹⁰ is 10x⁹ with no limit needed. Product, quotient and chain rules come next, and the chain rule gives the derivative of (3x + 1)⁴ in one line instead of expanding five terms. Rates of change turn height into velocity, and the last unit looks for slopes of zero to find the highest and lowest points. In the calculus lab you drag a point so that a secant shrinks into a tangent, and a purple dot on the dashed derivative curve sits at the height of the slope you are reading. With f(x) = x³ − 3x you can drag over the peak and the valley and watch the slope reach zero.
Before you start
The derivative is defined as a limit of secant slopes, so start with Limits. Comfort with exponents, expanding brackets and the slope of a line helps throughout, and the unit on product, quotient and chain rules also differentiates sine and cosine, so some familiarity with those functions helps.
When you finish
Afterwards you can find the derivative of x² from the difference quotient, read f′(x) = 2x as a slope for every x, and differentiate x¹⁰ with the power rule. You will use the product, quotient and chain rules, turn a height formula into velocity and acceleration for a dropped rock, and show that the 10 by 10 square has the largest area for perimeter 40.
- 1Defining the derivativeFor f(x) = x², the slope of the secant from x = 1 to x = 1 + h works out to exactly 2 + h. Let h shrink to nothing and 2 is all that is left.
- 2The derivative as a functionThe slope of y = x² is −6 at x = −3, 0 at the bottom, and 6 at x = 3. One formula, f′(x) = 2x, gives all of them.
- 3The power rule and basic rulesThe derivative of x¹⁰ is 10x⁹. Bring the exponent down in front and lower it by one; no limit needed.
- 4Product, quotient and chain rulesExpanding (3x + 1)⁴ takes five terms of algebra. The chain rule gives its derivative, 12(3x + 1)³, in one line.
- 5Derivatives as rates of changeA rock dropped from 64 feet has height −16t² + 64. Differentiate once and you get its velocity, −32t; it hits the ground at t = 2 moving at 64 feet per second.
- 6Maxima and minimaOf all rectangles with perimeter 40, the one with the largest area is the 10 by 10 square. The derivative finds it by looking for a slope of zero.
Sources for this course (10)
- OpenStax, Calculus Volume 1 : 2.1 A Preview of Calculus (tangent as the limit of secants; rock s(t) = −16t² + 64; stone h(t) = 15t − 4.9t² in the exercises: openstax.org/books/calculus-volume-1/pages/2-1-a-preview-of-calculus)
- chapter 3 Derivatives: 3.1 Defining the Derivative (difference quotient, f′(a) = lim (f(a + h) − f(a))/h, tangent line: .../pages/3-1-defining-the-derivative)
- 3.2 The Derivative as a Function (graphing f′, differentiability implies continuity, |x| has a corner at 0, higher derivatives: .../pages/3-2-the-derivative-as-a-function)
- 3.3 Differentiation Rules (constant, power, sum, difference, constant multiple, product and quotient rules: .../pages/3-3-differentiation-rules)
- 3.4 Derivatives as Rates of Change (position, velocity, speed, acceleration: .../pages/3-4-derivatives-as-rates-of-change)
- 3.5 Derivatives of Trigonometric Functions (d/dx sin x = cos x using lim (sin h)/h = 1 and lim (cos h − 1)/h = 0: .../pages/3-5-derivatives-of-trigonometric-functions)
- 3.6 The Chain Rule (.../pages/3-6-the-chain-rule)
- chapter 4: 4.3 Maxima and Minima (critical points, Fermat's theorem, extreme value theorem, closed interval method: .../pages/4-3-maxima-and-minima)
- 4.5 Derivatives and the Shape of a Graph (first derivative test: .../pages/4-5-derivatives-and-the-shape-of-a-graph)
- 5.3 for Newton and Leibniz working in the late 1600s and early 1700s (.../pages/5-3-the-fundamental-theorem-of-calculus).
Checked: 2026-10-17